On the number of zeros of nonoscillatory solutions to half-linear ordinary differential equations involving a parameter

Abstract:

In this paper the following half-linear ordinary differential equation is considered: \begin{equation} \tag {$\mathrm {H}_{\lambda }$} (|x’|^{\alpha }\operatorname {sgn} x’)’ + \lambda p(t)|x|^{\alpha }\operatorname {sgn} x =0,\qquad t \geq a, \end{equation} where $\alpha > 0$ is a constant, $\lambda > 0$ is a parameter, and $p(t)$ is a continuous function on $[a, \infty )$, $a > 0$, and $p(t) > 0$ for $t \in [a, \infty )$. The main purpose is to show that precise information about the number of zeros can be drawn for some special type of solutions $x(t; \lambda )$ of (H$_{\lambda })$ such that \[ \lim _{t\to \infty }\frac {x(t; \lambda )}{\sqrt {t}} = 0. \] It is shown that, if $\alpha \geq 1$ and if (H$_{\lambda })$ is strongly nonoscillatory, then there exists a sequence $\{\lambda _{n}\}_{n=1}^{\infty }$ such that $0=\lambda _{0}<\lambda _{1}<\cdots < \lambda _{n}<\cdots$, $\lambda _{n} \to +\infty$ as $n \to \infty$; and $x(t; \lambda )$ with $\lambda = \lambda _n$ has exactly $n-1$ zeros in the interval $(a,\infty )$ and $x(a; \lambda _n) = 0$; and $x(t; \lambda )$ with $\lambda \in (\lambda _{n-1}, \lambda _n)$ has exactly $n-1$ zeros in $(a,\infty )$ and $x(a; \lambda _n) \neq 0$. For the proof of the theorem, we make use of the generalized Prüfer transformation, which consists of the generalized sine and cosine functions.